On finite covers of surfaces with boundary…

I have a new preprint on the arxiv, joint with Emily Stark. We provide the first known examples of one-ended hyperbolic groups which are not abstractly coHopfian. That means that there is a one ended hyperbolic group \(G\) which contains a finite index subgroup \(G’ \leq G \) that embeds \( G’ \rightarrow G\) as an infinite index subgroup. I encourage you to look at the paper for details. The main example and proof can be drawn out on a single side of A4 — it’s a simple surface amalgam and we exploit the tremendous flexibility you have when you take a finite cover of a surface with boundary.

We use the following Lemma extensively. It’s from Walter Neumann’s 2001 paper Immersed and virtually embedded \(\pi_1\)-injective surfaces in 3-manifolds, although, as he says, it is apparently “well known”.

The utility of this Lemma is that it reassures you that if you can imagine your desired cover — such as the following I’ve drawn below — and it satisfies a basic necessary Euler characteristic computations, then the cover does in fact exist.

The surface \(F’\) is a degree 5 cover. The collection of degrees in this example are 1+3+1=5. Note that if the 3 was replaced with a 2 and I tried to find a degree 4 cover with the specified boundary it would be impossible.

For our main example you can compute your desired covers by hand, but it is worth knowing what kind of covers of a surface with boundary you can take. This lemma tells you exactly how much control you have. And mathematical research, like all forms of insecurity, is really all about control.

The proof given above is brief, to say the least, so I think it is worth expanding on the details.

First, we remind ourselves how you might construct such a cover by hand. Take a surface with genus one and a single boundary component. From a group theoretic point of view this is just the free group generated by two element \( \mathbb{F}_2 = \langle x, y \rangle \) bundled together with the conjugacy class of the commutator \( [x,y] \). For me at least, finding finite index subgroups of the free group boils down to futzing around with graphs. Thus, we let \(X\) be a bouquet of two circles and let \(\langle x,y \rangle = \pi_1X\).

I actually drew it by hand.

On the right I drew the surfaces with boundary and on the left I drew the corresponding graphs with the loop corresponding to the boundary. Once you have drawn the graphs out it’s easy to verify that you have the boundary components you want. The trick is knowing you can find the desired finite covers. The key insight is that \(\alpha\)-sheeted covers of a graph \(X\) are in a correspondence with representations into the permutation group on \(n\) elements: \( \pi_1 X \rightarrow \textrm{Sym}(\alpha) \)

We can see how this correspondence works in practice in the example I just drew:

Giving each of vertex a number we can see that the edges labelled by a given generator of our free group gives a permutation. In this example the generator \(x\) gives the permutation \((1,2)(3)\) (see the red edges on the right), while the generator \(y\) gives the permutation \((1,2,3)\) (see the blue edges on the left). Thus we have a homomorphism determined by mapping the generators to the corresponding permutation.

At this point we need to be careful because there are some left-right issues hidden here. When I multiply group elements \(xy\) I am composing paths in the fundamental group. That means I concatenate the corresponding paths, starting with the \(x\) path and then following it with the \(y\) path. I’m reading the composition from left to right. In contrast, when I usually compose a pair of permutations \(\sigma_1 \sigma_2\) I compute the composition by reading them from right to left. But in order to be able to interpret my homomorphism correctly I’m going to have to compose my permutations in reverse order, from left to right.

Now we can compute the image of the element corresponding to the boundary curve: $$ xyx^{-1}y^{-1} \mapsto (1,2) \circ (1,2,3) \circ (1,2) \circ (3,2,1) = (1,2,3).$$ Tracing out how the boundary curve lifts is equivalent to computing this permutation element. This makes it clear that there is a single boundary component covering the previous with degree 3. (In this example it doesn’t matter in which direction we composed the permutations).

Conversely, choosing pair of permutions, say, $$(1,2)(3,4)\textrm{, and } (2,4,3) \in \textrm{Sym(4)}$$ to be the images of \(x\) and \(y\) we can construct a corresponding cover by taking 4 vertices and adding the appropriate labelled edges:

Now when we compute (remembering to compose our permutations from left to right) the image of our commutator element we get $$ xyx^{-1}y^{-1} \mapsto (1,2)(3,4) \circ (2,4,3) \circ (1,2)(3,4) \circ (3,4,2) = (1,4)(2,3).$$ Thus the surface has two boundary components, each covering the boundary in the base surface with degree two.

The take away from this discussion is that finding suitable covers corresponds to finding a suitable homomorphism $$\phi : \pi_1 X \rightarrow \textrm{Sym($\alpha$)}$$ such that the image of the elements corresponding to boundary curves are permutaions with the desired decomposition into cycles. Our weapon of choice is the fact that any even permutation can be written as the commutator of an \(\alpha\)-cycle and an involution:

Un résultat extrémal en théorie des permutations. Jacques, Alain; Lenormand, Claude; Lentin, André; Perrot, Jean-François, C. R. Acad. Sci. Paris Sér. A-B 266 1968 A446–A448

First we consider the case where \(\Sigma\) has a single boundary component. So \(\Sigma\) is a surface with genus \(g\), Euler characteristic $$ \chi(\Sigma) =  2 – 2g – |\partial \Sigma | $$ and a single boundary component, so \(|\partial \Sigma | = 1\). This corresponds to the free group generated by \(2g\) elements and the group element corresponding to the boundary, which is the product of commutators: $$( \langle x_1, y_1, \ldots x_g, y_g \rangle, [x_1,y_1]\cdots [x_g, y_g] ).$$

Suppose we wish to construct a cover of degree \(\alpha\) with the boundary components of degrees \(\alpha_1, \ldots, \alpha_k\). Then apply the above Theorem to the permutation $$\sigma = (1, \ldots, \alpha_1)(\alpha_1 +1, \ldots, \alpha_1 + \alpha_2) \cdots (\alpha_1 + \cdots + \alpha_{k-1} +1, \ldots, \alpha).$$ The theorem only applies if \(\sigma\) is an even permutation, which we compute to be equivalent to $$ \sum_i (\alpha_i -1) = \alpha – k$$ being even. As \(\chi(\Sigma) = 1-2g\) this is equivalent to \(k\) having the same parity as \(\alpha\chi(\Sigma)\), the sufficient condition given in the statement of our theorem.

Thus there exists permutations \( \sigma_x, \sigma_y \in \textrm{Sym}(\alpha)\) such that \( [\sigma_x, \sigma_y] = \sigma \). The homomorphism $$ \phi: \pi_1 \Sigma \rightarrow \textrm{Sym($\alpha$)}$$ given by mapping \(x\) to \(\sigma_x\) and \(y\) to \(\sigma_y\) therefore corresponds to a cover with the desired boundary.

Now we consider the slightly trickier general case where we have multiple boundary components, which is to say \(|\partial \Sigma| = b\). In which case the pair \((\Sigma, \partial \Sigma)\) corresponds to $$ (\langle x_1, y_1, \ldots, x_g, y_g, t_1, \ldots t_{b-1} \rangle , \{t_1,\ldots, t_{b-1}, t_{b-1}\cdots t_{1}[x_1,y_1] \cdots[x_n,y_n] \} ).$$

Now suppose we desire that the \(i\)-th boundary component is covered with degrees \(\alpha_1^i, \ldots, \alpha_{k_i}^i\), then let $$ \sigma_i = (1, \ldots, \alpha_1^i)(\alpha_1^i +1, \ldots, \alpha_1^i + \alpha_2^i) \cdots (\alpha_1^i + \cdots + \alpha_{k_i-1}^i +1, \ldots, \alpha)$$ for \(1 \leq i \leq b\). Now we wish to find \(\sigma_x, \sigma_y\) such that $$[\sigma_x, \sigma_y] = \sigma_1 \cdots \sigma_b.$$ This requires that the product of the \(\sigma_i\) is even. This means that the sum $$ \sum_i \sum_j (\alpha_j^i – 1) = \sum_i (\alpha – k_i) =\alpha b – \sum_i k_i = \alpha (\chi(\Sigma) -2 + 2g) – \sum_i k_i$$ should be even, which is true precisely when the total number of prescribed boundary components \(\sum_i k_i\) has the same parity as \(\alpha \chi(\Sigma)\).

Given that our parity condition is satisfied, we define our homormorphism \(\pi_1 \Sigma \rightarrow \textrm{Sym}(\alpha)\) as follows:

$$\begin{align} x_1 \mapsto & \sigma_x  \\ y_1 \mapsto & \sigma_y \\ x_2 \mapsto & 1 \\ \vdots \\ y_g \mapsto & 1 \\ t_1 \mapsto & \sigma_1^{-1} \\ t_2 \mapsto & \sigma_2^{-1} \\ \vdots \\ t_{b-1} \mapsto & \sigma_{b-1}^{-1} \end{align} $$

Then it only remains to verify that $$ t_{b-1}\cdots t_{1} [x_1,x_2] \cdots [x_g, y_g] \mapsto \sigma_{b-1}^{-1} \cdots \sigma_1^{-1} [\sigma_x, \sigma_y] =  \sigma_{b-1}^{-1} \cdots \sigma_1^{-1} \sigma_1 \cdots \sigma_{b-1} \sigma_b = \sigma_b $$ and conclude this gives us our desired cover.

QED.

(I’d like to thank Emily for informing me about Neumann’s Lemma, and Nir for various discussions related to this.)

Chicago and back to Haifa

I finally managed to capture the chalkboard on camera. The diagram is the central focus and remains up through the entire talk, with various additions and modifications made along the way.

I like to commit to the diagram.

Giving the same talk several times over the space of a month allows you to appreciate a few things. Minor adjustments can make a big difference, especially to pacing. Having already given a talk once really boosts your confidence. Having given a talk four times already doesn’t give you complete confidence. Responses can vary dramatically as audiences latch onto different aspects of what you are doing, and as you invariably emphasize specific things. At Boston College they wanted to hear more about special cube complexes, something many of them had heard about but had little exposure to, while at UIC I had audience members who had themselves considered the specific conjecture I am working on and were curious to hear about the obstructions I had encountered.

After roaming all over Manhattan I had a far more limited Chicago experience. I don’t think I strayed any further than a mile from the hotel and campus and most of my time was spent talking mathematics.

Now I’m back in Haifa, recovering from jet-lag. Despite the rain that marked my departure a month ago, I’ve returned to amazing weather.

Boston and then New York

After a week in Boston I’m now in New York. I was out there just over a week and gave seminar talks at Boston College and Tufts. I also saw a free performance of Hamlet at Wellesley College given by Actors From the London Stage. I studied Hamlet for AS-level English literature, so at least ten years ago I had more than a passing familiarity with the play. Although I watched multiple film adaptions at the time (including Hamlet 2000 which I believe was widely considered disappointing, but has a stellar cast and may be worth revisiting), I don’t believe I ever actually saw it on stage.

It was a pleasant suprise to discover that Hamlet is great. The initial palace intrigue really does work, once you have taken a ten-year-step-back from studying it on the page. It made me yearn for ghosts to return from their horror fiction exile and claim their rightful place in contemporary drama.

I also went for a walk in the woods and met this fellow:

Now I am in New York. The Big Apple, where everything is Big expensive, and I’ve been going on Big walks, and now my Big legs feel like they’re going to fall off. Oh. And everything is Big expensive. It’s like I never even left the airport.

It’s like London with a grid system, less rain, and museums that aren’t free.

I’ve been having a blast here.

On top of my list of things I was excited to do in NY was visit The Strand bookstore. I miss the joys of English language bookstores while living in Israel, and The Strand is the epitome of everything I wanted. Extensive and well curated stacks brimming with wonderful literature. The store’s reputation came to me partly through word of mouth, but also through youtube videos I subsequently stumbled upon. For example:

Infuriatingly, the video doesn’t seriously set out to answer the question posed. I still found it engrossing.

(The individual featured in this video, Michael Orthofer, is a fascinating figure. He is the editor-and-contributor-in-chief of The Complete Review, an internet passion project dedicated to reading, reviewing and documenting the most significant works of contemporary literature. And I don’t mean just English literature. I mean world literature. The guy is reading upwards of five books a week and reads in multiple languages. He writes on the blog how much of a relief it was that  no Nobel prize for literature was awarded this year, leaving him free to dedicate more time to the actual business of reading and writing. The New Yorker ran a profile on him and his obsession. If you are bookishly inclined you may have already stumbled across his website in the past. It is old school, homegrown html at its finest. Take a look at it. It probably won’t surprise you that Orthofer wrote the website in the previous century. I ran into it while trying track down a translation of The Pornographers by Nozaka Akiyuki [I don’t even remember why].  It is hard not to already look on the website with a kind of nostalgia. It seems so far removed from what the internet has become. The whole concept is so insane, non-commercial, personal, and unscalable that it feels like a product of a bygone era. Even our academic webpages are now becoming slick, Web 2.0, bland, templated monstrosities. It is interesting to note, however, that The Complete Review outlasted many of its contemporaries that were more commercially inclined. Like Pets.com.)

Although I am determined to travel light on my north American excursion, I went a little nuts and bought a piles of books. As with all the most dreadfully insane acts, it was of a deeply premeditated nature, with it’s own internal logic and self justification. I bought a bunch of books, but only short books.

The Scarlet Letter I bought in Boston, the volumes above in Montreal, and everything below in New York.

I also spent a day in the Met, went to see an adaption of Vonnegut’s Mother Night at 59E59, dropped by New York Public Library to see Winnie the Pooh, walked across the Brooklyn bridge, and visited ground zero.

My seminar talk at Columbia went fantastically well. I’ve failed to take a picture of the chalkboard after any of the seminar talks I’ve given so far, but there is still UIC. And I’ll probably scan or photograph my notes as well.

Columbia’s campus is amazing.

As my trip progresses and my final return flight approaches, so Halloween also approaches.

Some Halloween treats.

Returning to Montreal

The first overcast sky I’d seen in quite a while.

I set off from Haifa just as the first rain since the start of the summer started to fall. On arrival in Montreal the shorts and T-shirt I had set off in became insufficient to the demands of the local climate, so I had to switch into warmer clothes and revive myself with Tim Horton’s hot chocolate.

Two years had passed since I was last in Montreal to defend my thesis — the morning after the 2016 election. It was interesting to see what had changed. Cannabis was about to be legalized. The building work on McGill/Sherbrooke had been completed. The math department was full of unfamiliar faces. Many restaurants had closed (no more SmartBurger). Murals commemorating the late Leonard Cohen had appeared across the city. Poutine was pretty much as I remember it, though.

Montreal is now also home to the world’s largest permanent barbie exhibit. Which is completely free to access.

One among many of the Barbies on display.

Canadian Thanksgiving took us all by surprise, so I gave my McGill seminar talk earlier than expected. It was the first outing of the talk I’m touring while I’m out here in North America, presenting my new proof (and generalization) of Leighton’s graph covering theorem. The seminar went well, but I feel like I’m going to be experimenting a little to find the best way to explicate the coset summation argument I use. Piotr appreciated the fact that I was able to give the entire proof within the hour.

I was also able to do quite a bit of math. A fair chunk of Thanksgiving was spent discussing math with Dani and scrawling our ideas all over my notebook.

Don’t read too much into anything going on here…

I also took pictures of new murals I spotted around this city:

Gromov, cheese, pretending to quit mathematics, and French.

In December last year, the Notices of the AMS ran a collection of reminiscences in memory of Marcel Burger (1927-2016), the late French differential geometer. He was also a former director of the Institut des Haute Etudes Scientific and, according to the Wikipedia, played a major role in getting Gromov positions in Paris and at the IHES in the 80s. Gromov contributed to the article, listing Berger’s mathematical achievements, before sharing a more personal anecdote:

Within my own field Gromov has had a profound influence. His essay Hyperbolic Groups led to the term “combinatorial group theory” being more or less abandoned and replaced with “geometric group theory”. As a graduate student I found the monograph frequently cited as the origin for an astounding range of ideas. At some point I had trouble finding a copy of the paper online and for a brief moment wondered if the paper itself were just an urban myth or elaborate hoax.

Gromov’s foray into group theory is just one episode in a long career. His first major breakthrough was in partial differential equations; the “h-principal” which, according to Larry Guth, was analogous to observing that you don’t need to give an explicit description of how to put a wool sweater into a box in order to know that you can actually put it into the box. It is actually a little hard, at least for myself, to get a full grasp on Gromov’s other contributions as they span unfamiliar fields of mathematics, but I recommend this nice What is… article, written by the late Marcel Berger, describing Gromov’s contribution to the understanding of isosystolic inequalities.

There is the perception about great mathematicians that, while they are no doubt very clever, somehow they have lost a little of the common sense that the rest of us possess. I’m naturally inclined to discount such thinking; I am far happier believing that we are all fool enough to take absence of common sense, on certain occasions. However, it is hard to dismiss the  idea entirely given the following admission by Gromov, in his personal autobiographical recollections he wrote on receipt of the Abel prize in 2009:

The passage speaks for itself, but I wish to emphasize that Gromov’s discovery of the correct pronunciation of French verb endings came after ten years of living in Paris. You don’t have to have extensive experience learning foreign languages to appreciate how remarkable an oversight this is. It certainly puts his remarks in other interviews about dedicating one’s life to mathematical pursuits in a rather strong light.

If you read the entire autobiographical essay you will find it rather short on biography. The best biographical details I have found came from this La Monde article written on his being awarded the Abel prize. Even then the details seems to be coming second hand. The most interesting parts concern his leaving the Soviet Union:

I wanted to leave the Soviet Union from the age of 14. […] I could not stand the country. The political pressure there was very unpleasant, and it did not come only from the top. […] The professors had to teach in such a way as to show respect for the regime. We felt the pressure of always having to express our submission to the system. One could not do that without deforming one’s personality and each mathematician that I knew ended up, at a certain age, developing a neurosis accompanied  by severe disorders. In my opinion, they had become sick. I did not want to reach that point.

Gromov, according to Georges Ripka, via La Monde (apologies for my translation)

As the article goes on to explain, Gromov decided his best chance to escape was to hide his mathematical talent. He quit math, quit his university, and burned all his academic bridges. He stopped producing mathematics. Or at least writing it. He joined some meteorological institute and did research on paper pulp. Eventually, he was granted permission to emigrate to Israel, but on landing in Rome in 1974 he set off for the States instead, where Jim Simons secured him a position at Stony Brook.

(As a side note, this is Jim Simons of Renaissance Technology and Simons foundations fame. Simons, aside from his mathematical contributions, is probably one of the most important mathematicians alive in terms of funding, supporting, and propagandizing for mathematics. He is considered influential enough for the New Yorker to profile. Alongside Gromov, he is one of the names that every mathematicians should know.)

Yom Kippur, 2018

Yom Kippur, or the Day of Atonement, is the tenth day in the Jewish new year. It precedes the start of the new academic year here in Israel. Due to all the High Holidays — Rosh Hashanah, Yom Kippur, and Sukkot — new arrivals at the Technion, eager to start postdoctoral fellowships, find themselves sitting things out for a month until university life commences. For those of us already here, this is usually time spent getting job application material together. So it isn’t the most exciting month of the year for us.

There is a silver lining: witnessing first hand the one day (from sunset to sunset) that the streets of Israel are empty of cars. Because Yom Kippur is more than just a legal holiday. Everything is shut down: TV, radio, airports, shops, and public transport. I’ve even heard a third hand story of someone with a minor injury being told to wait until the next day before going in to hospital.

I was in Israel two years ago for Yom Kippur. I went out for an evening jog and, to my pleasant surprise, found the streets full of children tearing downhill on their push bikes being chased by their parents. While observant Israeli’s fast, a lot of secular Israelis — kids in particular — take the opportunity to do some serious cycling. You need to appreciate how unfriendly Israeli roads are to cyclists every other day of the year to understand how magical a time this must be.

This year we went out for a walk around the Carmel center just after sunset to enjoy the empty streets. We spotted this car sitting in the middle of the street, apparently abandoned as the final seconds of the day had passed by.

 You can find plenty of videos on youtube of people out cycling on the empty streets, but here is just one of them:

As you can see from the video, it is inaccurate to say that there are absolutely no cars on the road. In this video, at least, the reason is that their route passes through Daliyat al-Karmel , a Druze town, where few people observe the holiday.

Even within Haifa, however, we were still able to spot a few cars on the streets. One in particular had a gang of skateboarders clinging to it. I was just quick enough to take the following picture:

This year I wanted to see the highway deserted, so in the afternoon we walked down from the mountain to the beach, crossing the main road along the coast on the way.

One way or another, this was my last Yom Kippur in Israel. I think it might be the one facet of Israeli life — a day when even the cars stop — that I will miss the most.